Optimal. Leaf size=26 \[ -\frac {1}{3} e^{-3 x}-\frac {e^{-2 x}}{2}-e^{-x} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2320, 14}
\begin {gather*} -\frac {1}{3} e^{-3 x}-\frac {e^{-2 x}}{2}-e^{-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2320
Rubi steps
\begin {align*} \int e^{-4 x} \left (e^x+e^{2 x}+e^{3 x}\right ) \, dx &=\text {Subst}\left (\int \frac {1+x+x^2}{x^4} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{x^4}+\frac {1}{x^3}+\frac {1}{x^2}\right ) \, dx,x,e^x\right )\\ &=-\frac {1}{3} e^{-3 x}-\frac {e^{-2 x}}{2}-e^{-x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{6} e^{-3 x} \left (-2-3 e^x-6 e^{2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.77
method | result | size |
default | \(-\frac {{\mathrm e}^{-3 x}}{3}-\frac {{\mathrm e}^{-2 x}}{2}-{\mathrm e}^{-x}\) | \(20\) |
risch | \(-\frac {{\mathrm e}^{-3 x}}{3}-\frac {{\mathrm e}^{-2 x}}{2}-{\mathrm e}^{-x}\) | \(20\) |
meijerg | \(\frac {11}{6}-\frac {{\mathrm e}^{-3 x}}{3}-\frac {{\mathrm e}^{-2 x}}{2}-{\mathrm e}^{-x}\) | \(21\) |
norman | \(\left (-\frac {{\mathrm e}^{2 x}}{2}-{\mathrm e}^{3 x}-\frac {{\mathrm e}^{x}}{3}\right ) {\mathrm e}^{-4 x}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 19, normalized size = 0.73 \begin {gather*} -e^{\left (-x\right )} - \frac {1}{2} \, e^{\left (-2 \, x\right )} - \frac {1}{3} \, e^{\left (-3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 18, normalized size = 0.69 \begin {gather*} -\frac {1}{6} \, {\left (6 \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2\right )} e^{\left (-3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 22, normalized size = 0.85 \begin {gather*} - e^{- x} - \frac {e^{- 2 x}}{2} - \frac {e^{- 3 x}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.79, size = 18, normalized size = 0.69 \begin {gather*} -\frac {1}{6} \, {\left (6 \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2\right )} e^{\left (-3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 18, normalized size = 0.69 \begin {gather*} -\frac {{\mathrm {e}}^{-3\,x}\,\left (6\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+2\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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